mlsanityData to 14 Oct 2026

Geodesic Exponential Kernels: When Curvature and Linearity Conflict

arXiv 1411.0296 ↗First posted 2 Nov 2014cs.LG cs.CVcited by 47

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Abstract (arXiv)

We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive definite kernel on a geodesic metric space if the space is flat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive definite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verified empirically.

Key facts. Cited by 47 papers in the map (data through 14 Oct 2026). First posted on arXiv 2 Nov 2014.

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Citing papers per year: 2021: 8; 2022: 10; 2023: 11; 2024: 5; 2025: 6; 2026: 7.

47 papers.

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  1. 2021Generalized Shape Metrics on Neural Representationscited by 112
  2. 2021Evaluation Metrics for Graph Generative Models: Problems, Pitfalls, and Practical Solutionscited by 39
  3. 2022Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact casecited by 25
  4. 2021Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernelscited by 24
  5. 2024A survey and benchmark of high-dimensional Bayesian optimization of discrete sequencescited by 23
  6. 2021Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matricescited by 22
  7. 2021Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernelscited by 19
  8. 2023Curvature Filtrations for Graph Generative Model Evaluationcited by 19
  9. 2022Distribution Regression with Sliced Wasserstein Kernelscited by 18
  10. 2023Metric Space Magnitude for Evaluating the Diversity of Latent Representationscited by 17
  11. 2022Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projectionscited by 16
  12. 2023Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Spacecited by 11
35 more
  1. 2024The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphscited by 11
  2. 2022Crosslinguistic word order variation reflects evolutionary pressures of dependency and information localitycited by 10
  3. 2021Dimension Reduction for Fr'echet Regressioncited by 8
  4. 2024Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variatescited by 8
  5. 2026Kernel-Gradient Drifting Modelscited by 7
  6. 2022Boundary-Aware Uncertainty for Feature Attribution Explainerscited by 6
  7. 2022Optimization on Manifolds via Graph Gaussian Processescited by 6
  8. 2023Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metricscited by 6
  9. 2023L3DMC: Lifelong Learning using Distillation via Mixed-Curvature Spacecited by 6
  10. 2023Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approachcited by 6
  11. 2023The Gaussian kernel on the circle and spaces that admit isometric embeddings of the circlecited by 5
  12. 2021Marginalising over Stationary Kernels with Bayesian Quadraturecited by 4
  13. 2023Implicit Manifold Gaussian Process Regressioncited by 4
  14. 2025Modeling Hierarchical Spaces: A Review and Unified Framework for Surrogate-Based Architecture Designcited by 4
  15. 2022Time-inhomogeneous diffusion geometry and topologycited by 3
  16. 2023Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definitecited by 3
  17. 2023Geometric Learning with Positively Decomposable Kernelscited by 3
  18. 2022Self-supervised Representation Learning on Electronic Health Records with Graph Kernel Infomaxcited by 2
  19. 2022Gaussian Process regression over discrete probability measures: on the non-stationarity relation between Euclidean and Wasserstein Squared Exponential Kernelscited by 2
  20. 2023Gaussian Process on the Product of Directional Manifoldscited by 2
  21. 2025Scalable Geometric Learning with Correlation-Based Functional Brain Networkscited by 2
  22. 2025Universal kernels via harmonic analysis on Riemannian symmetric spacescited by 2
  23. 2021Geodesic squared exponential kernel for non-rigid shape registrationcited by 1
  24. 2024Learning from Summarized Data: Gaussian Process Regression with Sample Quasi-Likelihoodcited by 1
  25. 2025Keep your distance: learning dispersed embeddings on mathbbSmcited by 1
  26. 2022Hyperbolic Centroid Calculations for Text Classificationcited by 0
  27. 2024Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimationcited by 0
  28. 2025A probabilistic view on Riemannian machine learning models for SPD matricescited by 0
  29. 2025Categorical and geometric methods in statistical, manifold, and machine learningcited by 0
  30. 2026Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approachcited by 0
  31. 2026Information Theoretic Bayesian Optimization over the Probability Simplexcited by 0
  32. 2026Heat and Mat'ern Kernels on Matchingscited by 0
  33. 2026Intrinsic effective sample size for manifold-valued Markov chain Monte Carlo via kernel discrepancycited by 0
  34. 2026A General Kernel Framework for Non-CND Distance Measures Using |D|-Dimensional Sparse Landmark Embeddingscited by 0
  35. 2026RW-Flow: One-Step Generation on Compact Manifolds via Wasserstein Gradient Flowscited by 0